In this paper, we establish the boundedness estimates for the composition of the homotopy operator $T$ and the potential operator $T_{\Phi }$ on differential forms with Orlicz–Lipschitz norm and Orlicz-BMO norm which are defined by a Young function. Moreover, we derive the two-weight norm inequalities for the composite operator $T\circ T_{\Phi }$ using the Poincaré-type inequality with $A_r^{\lambda }(\Omega )$-weight. Finally, we demonstrate some applications of our main results.

In this article we show some density properties of smooth and compactly supported functions in fractional Sobolev spaces with variable exponents. The additional difficulty in this nonlocal setting is caused by the fact that the variable exponent Lebesgue spaces are not translation-invariant.

We prove that every bijective map that preserves mixed Lie triple products from a factor von Neumann algebra $\mathcal{M}$ with $dim\mathcal{M}>4$ into another factor von Neumann algebra $\mathcal{N}$ is of the form $A\to ϵ\Psi \left(A\right)$, where $ϵ\in \{1,-1\}$ and $\Psi :\mathcal{M}\to \mathcal{N}$ is a linear $\ast$-isomorphism or a conjugate linear $\ast$-isomorphism. Also, we give the structure of this map when $dim\mathcal{M}=4$.

Let $(\mathcal{X},d,\mu )$ be a metric measure space such that, for any fixed $x\in \mathcal{X}$, $\mu \left(B\right(x,r\left)\right)$ is a continuous function with respect to $r\in (0,\infty )$. In this paper, we prove endpoint estimates for the multilinear fractional integral operators $I_{m,\alpha }$ from the product of Lebesgue spaces $L^1\left(\mu \right)\times \cdots \times L^1\left(\mu \right)\times L^{p_{k+1}}\left(\mu \right)\times \cdots \times L^{p_m}\left(\mu \right)$ into the Lebesgue space $L^q\left(\mu \right)$, where $k\in [1,m)\cap \mathbb{N}$, $\alpha \in [k,m)$, $p_i\in (1,\infty )$ for $i\in \{k+1,\dots ,m\}$ and $1/q=k+{\sum }_{i=k+1}^{m}1/p_i-\alpha$. We furthermore prove that $I_{m,\alpha }$ is bounded from $L^{p_1}\left(\mu \right)\times \cdots \times L^{p_m}\left(\mu \right)$ into $L^{\infty }\left(\mu \right)$, where $p_i\in (1,\infty )$ for $i\in \{1,\dots ,m\}$ and ${\sum }_{i=1}^{m}1/p_i=\alpha \in [1,m)$.

An operator $T\in \mathcal{B}\left(\mathcal{H}\right)$ is complex-symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\to \mathcal{H}$ so that $CTC=T^{\ast }$. In this note, we prove that on finite-dimensional Hilbert space ${\mathbb{C}}^n$ with $n\ge 3$, noncomplex symmetric operators are dense in $\mathcal{B}\left({\mathbb{C}}^n\right)$.

In this paper, we give some refinements and reverses for the Hermite–Hadamard inequality when the integrand map is the Fenchel conjugate in convex analysis. The theoretical results obtained by our present functional approach immediately imply those of an operator version in a simple and elegant way. An application for scalar means is provided as well.

Let $(G,G_{+})$ be a lattice-ordered abelian group with positive cone $G_{+}$, and let $H_{+}$ be a hereditary subsemigroup of $G_{+}$. In previous work, the author and Pryde introduced a closed ideal $I_{H_{+}}$ of the $C^{\ast }$-subalgebra $B_{G_{+}}$ of ${\ell }^{\infty }\left(G_{+}\right)$ spanned by the functions $\{1_x:x\in G_{+}\}$. Then we showed that the crossed product $C^{\ast }$-algebra $B_{(G/H{)}_{+}}{\times }_{\beta }{G}_{+}$ is realized as an induced $C^{\ast }$-algebra ${Ind}_{H^{\perp }}^{\hat{G}}\left(B_{(G/H{)}_{+}}{\times }_{\tau }\right(G/H{)}_{+})$. In this paper, we prove the existence of the following short exact sequence of $C^{\ast }$-algebras: $$0\to I_{H_{+}}{\times }_{\alpha }{G}_{+}\to B_{G_{+}}{\times }_{\alpha }{G}_{+}\to {Ind}_{H^{\perp }}^{\hat{G}}\left(B_{(G/H{)}_{+}}{\times }_{\tau }\right(G/H{)}_{+})\to 0.$$ This relates $B_{G_{+}}{\times }_{\alpha }{G}_{+}$ to the structure of $I_{H_{+}}{\times }_{\alpha }{G}_{+}$ and $B_{(G/H{)}_{+}}{\times }_{\beta }{G}_{+}$. We then show that there is an isomorphism $\iota$ of $B_{H_{+}}{\times }_{\alpha }{H}_{+}$ into $B_{G_{+}}{\times }_{\alpha }{G}_{+}$. This leads to nontrivial results on commutator ideals in $C^{\ast }$-crossed products by hereditary subsemigroups involving an extension of previous results by Adji, Raeburn, and Rosjanuardi.

Fix $N\in \mathbb{N}$, and assume that, for every $n\in \{1,\dots ,N\}$, the functions $f_n:[0,1]\to [0,1]$ and $g_n:[0,1]\to \mathbb{R}$ are Lebesgue-measurable, $f_n$ is almost everywhere approximately differentiable with $|g_n(x)|<|f{\text{'}}_n(x)|$ for almost all $x\in [0,1]$, there exists $K\in \mathbb{N}$ such that the set $\{x\in [0,1]:card{f}_n^{-1}(x)>K\}$ is of Lebesgue measure zero, $f_n$ satisfy Luzin’s condition N, and the set $f_n^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset \mathbb{R}$ of Lebesgue measure zero. We show that the formula $Ph={\sum }_{n=1}^N{g}_n\cdot (h\circ f_n)$ defines a linear and continuous operator $P:L^1([0,1])\to L^1([0,1])$, and then we obtain results on the existence and uniqueness of solutions $\phi \in L^1([0,1])$ of the equation $\phi =P\phi +g$ with a given $g\in L^1([0,1])$.

We investigate the solvability for an infinite system of fractional order boundary value problems of differential equations in Banach sequence spaces ${\ell }_{\infty }$ and $\mathbf{c}$. Our approach depends on Darbo’s fixed point theorem in conjunction with new measures of noncompactness in spaces $C^1(J,{\ell }_{\infty })$ and $C^1(J,\mathbf{c})$.

Let be a closed operator matrix acting in the Hilbert space $\mathcal{H}\oplus \mathcal{K}$. In this paper, we concern ourselves with the completion problems of $M_C$. That is, we exactly describe the sets ${\bigcup }_{C\in {\mathcal{C}}_B^{+}(\mathcal{K},\mathcal{H})}{\sigma }_{\ast }(M_C)$ and ${\bigcap }_{C\in {\mathcal{C}}_B^{+}(\mathcal{K},\mathcal{H})}{\sigma }_{\mathrm{cr}}(M_C)$, where ${\sigma }_{\ast }(M_C)$ includes the residual spectrum, the continuous spectrum, and the closed range spectrum of $M_C$, and ${\mathcal{C}}_B^{+}(\mathcal{K},\mathcal{H})$ denotes the set of closable operators $C:\mathcal{D}(C)\subset \mathcal{K}\to \mathcal{H}$ such that $\mathcal{D}(C)\supset \mathcal{D}(B)$ for a given closed operator $B$ acting in $\mathcal{K}$.

We extend inequalities for operator monotone and operator convex functions onto elements of the extended positive part of a von Neumann algebra. In particular, this provides an opportunity to extend the inequalities onto unbounded positive self-adjoint operators.

In the following we generalize the concept of Birkhoff–James orthogonality of operators on a Hilbert space when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T{\perp }_A^{B}S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying $\Vert T+\gamma S{\Vert }_A\ge \Vert T{\Vert }_A$ for all $\gamma \in \mathbb{C}$. We extend a theorem due to Bhatia and Šemrl by proving that $T{\perp }_A^{B}S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\mathcal{H}$ such that ${lim\hspace{0.17em}}_{n\to +\infty }\Vert T{x}_n{\Vert }_A=\Vert T{\Vert }_A$ and ${lim\hspace{0.17em}}_{n\to +\infty }\langle T{x}_n,S{x}_n{\rangle }_A=0$. In addition, we give some $A$-distance formulas. Particularly, we prove

$${inf\hspace{0.17em}}_{\gamma \in \mathbb{C}}\Vert T+\gamma S{\Vert }_A=sup\hspace{0.17em}\left\{\right|\langle Tx,y{\rangle }_A|;\Vert x{\Vert }_A=\Vert y{\Vert }_A=1,\langle Sx,y{\rangle }_A=0\}.$$ Some other related results are also discussed.